Answer: [tex]SA=18\sqrt{91}+54\sqrt{3}[/tex]
Step-by-step explanation:
You must apply the following formula for calculate the total surface area the regular pyramid:
[tex]SA=\frac{pl}{2}+B[/tex]
Where p is the perimeter of the base, l is the slant height and B is the area of the base.
Find the apothem of the hexagonal base:
[tex]a=\frac{s}{2tan(\frac{180}{n})}[/tex]
Where s is the side length and n is the number of sides the polygon.
Then:
[tex]a=\frac{6}{2tan(\frac{180}{6})}[/tex]
[tex]a=3\sqrt{3}[/tex]
Apply the Pythagorean Theorem to find the slant height:
[tex]l=\sqrt{(3\sqrt{3})^2+8^2}=\sqrt{91}[/tex]
The perimeter is:
[tex]p=s*6=6*6=36[/tex]
The area of the base is:
[tex]B=\frac{3\sqrt{3}*s^2}{2}[/tex]
Where s is the side length.
Then:
[tex]B=s^2(\frac{3\sqrt{3}}{2})=54\sqrt{3}[/tex]
Susbtituting values, you obtain:
[tex]SA=\frac{36*\sqrt{91}}{2}+54\sqrt{3}[/tex]
[tex]SA=18\sqrt{91}+54\sqrt{3}[/tex]
